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PRL 110, 156402 (2013)
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PHYSICAL REVIEW LETTERS
Phonon-Induced Spin-Spin Interactions in Diamond Nanostructures:
Application to Spin Squeezing
S. D. Bennett,1 N. Y. Yao,1 J. Otterbach,1 P. Zoller,2,3 P. Rabl,4 and M. D. Lukin1
1
Physics Department, Harvard University, Cambridge, Massachusetts 02138, USA
Institute for Quantum Optics and Quantum Information, Austrian Academy of Sciences, 6020 Innsbruck, Austria
3
Institute for Theoretical Physics, University of Innsbruck, 6020 Innsbruck, Austria
4
Institute of Atomic and Subatomic Physics, TU Wien, Stadionallee 2, 1020 Wien, Austria
(Received 21 January 2013; published 9 April 2013)
2
We propose and analyze a novel mechanism for long-range spin-spin interactions in diamond
nanostructures. The interactions between electronic spins, associated with nitrogen-vacancy centers in
diamond, are mediated by their coupling via strain to the vibrational mode of a diamond mechanical
nanoresonator. This coupling results in phonon-mediated effective spin-spin interactions that can be used
to generate squeezed states of a spin ensemble. We show that spin dephasing and relaxation can be largely
suppressed, allowing for substantial spin squeezing under realistic experimental conditions. Our approach
has implications for spin-ensemble magnetometry, as well as phonon-mediated quantum information
processing with spin qubits.
DOI: 10.1103/PhysRevLett.110.156402
PACS numbers: 71.55.i, 07.10.Cm, 42.50.Dv
Electronic spins associated with nitrogen-vacancy (NV)
centers in diamond exhibit long coherence times and
optical addressability, motivating extensive research on
NV-based quantum information and sensing applications.
Recent experiments have demonstrated coupling of NV
electronic spins to nuclear spins [1,2], entanglement with
photons [3], as well as single spin [4,5] and ensemble [6,7]
magnetometry. An outstanding challenge is the realization
of controlled interactions between several NV centers,
required for quantum gates or to generate entangled
spin states for quantum-enhanced sensing. One approach
toward this goal is to couple NV centers to a resonant
optical [8,9] or mechanical [10–12] mode; this is particularly appealing in light of rapid progress in the fabrication
of diamond nanostructures with improved optical and
mechanical properties [13–17].
In this Letter, we describe a new approach for effective
spin-spin interactions between NV centers based on
strain-induced coupling to a vibrational mode of a diamond
resonator. We consider an ensemble of NV centers
embedded in a single crystal diamond nanobeam, as
depicted in Fig. 1(a). When the beam flexes, it strains the
diamond lattice, which in turn couples directly to the spin
triplet states in the NV electronic ground state [18,19]. For
a thin beam of length L 1 m, this strain-induced spinphonon coupling can allow for coherent effective spin-spin
interactions mediated by virtual phonons. Based on these
effective interactions, we explore the possibility of generating spin squeezing of an NV ensemble embedded in the
nanobeam. We account for spin dephasing and mechanical
dissipation and describe how spin echo techniques and
mechanical driving can be used to suppress the dominant
decoherence processes while preserving the coherent
spin-spin interactions. Using these techniques we find
0031-9007=13=110(15)=156402(5)
that significant spin squeezing can be achieved with realistic experimental parameters. Our results have implications for NV ensemble magnetometry and provide a new
route toward controlled long-range spin-spin interactions.
Model.—The electronic ground state of the negatively
charged NV center is a spin S ¼ 1 triplet with spin states
labeled by jms ¼ 0; 1i as shown in Fig. 1(b). In the
presence of external electric and magnetic fields E~ and
~ the Hamiltonian for a single NV is (@ ¼ 1) [19]
B,
HNV ¼ ðD0 þ dk Ez ÞS2z þ B gs S~ B~
d? ½Ex ðSx Sy þ Sy Sx Þ þ Ey ðS2x S2y Þ;
NV centers
(a)
(1)
(c)
d
on
am
di
L
(b)
(d)
FIG. 1 (color online). (a) All-diamond doubly clamped mechanical resonator with an ensemble of embedded NV centers.
(b) Spin triplet states of the NV electronic ground state. Local
perpendicular strain induced by beam bending mixes the j 1i
states. (c) A collection spins in the two-level subspace fjþ1i;
j1ig is off-resonantly coupled to a common mechanical mode
giving rise to effective spin-spin interactions. (d) Squeezing of
the spin uncertainty distribution of an NV ensemble.
156402-1
Ó 2013 American Physical Society
PRL 110, 156402 (2013)
PHYSICAL REVIEW LETTERS
where D0 =2 ’ 2:88 GHz is the zero field splitting,
gs ’ 2, B is the Bohr magneton, and dk (d? ) is the ground
state electric dipole moment in the direction parallel
(perpendicular) to the NV axis [20,21].
Motion of the diamond nanoresonator changes the local
strain at the position of the NV center, which results in an
effective, strain-induced electric field [19]. We are interested in the near-resonant coupling of a single resonant
mode of the nanobeam to the j 1i transition of the NV,
with Zeeman splitting B ¼ gs B Bz =@, as shown in
Figs. 1(b) and 1(c). The perpendicular component of strain
E? mixes the j 1i states. For small beam displacements,
the strain is linear in its position and we write E? ¼
E0 ða þ ay Þ, where a is the destruction operator of the
resonant mechanical mode of frequency !m , and E0 is
the perpendicular strain resulting from the zero point
motion of the beam. We note that the parallel component
of strain shifts both states j 1i relative to j0i [22];
however, with near-resonant coupling ¼B !m D0
and preparation in the j 1i subspace, the state j0i remains
unpopulated and parallel strain plays no role in what
follows. Within this two-level subspace, the interaction
y of each NV is Hi ¼ gðþ
i a þ a i Þ, where i ¼
j 1ii h1j is the Pauli operator of the ith NV center
and g is the single phonon coupling strength. For many
NV centers
we introduce collective spin operators,
P
P
Jz ¼ 12 i j1ii h1jj1ii h1j and J ¼ Jx iJy ¼ i i ,
which satisfy the usual angular momentum commutation
relations. The total system Hamiltonian can then be
written as
H ¼ !m ay a þ B Jz þ gðay J þ aJþ Þ;
(2)
which describes a Tavis-Cummings type interaction
between an ensemble of spins and a single mechanical
mode [23]. In Eq. (2) we have assumed uniform coupling
of each spin to the mechanical mode for simplicity; in
general the coupling may be nonuniform. We also assume
that the NVs are sufficiently far apart so that we may safely
ignore direct dipole-dipole interactions between the spins.
We discuss these points further below.
To estimate the coupling strength g, we calculate the
strain for a given mechanical mode and use the experimentally obtained stress coupling of 0:03 Hz Pa1 in the NV
ground state [24,25]. We take a doubly clamped diamond
beam [see Fig. 1(a)] with dimensions L w, h such that
Euler-Bernoulli thin beam elasticity theory is valid [26].
For NV centers located near the surface of the beam we
obtain [24]
1=2
g
@
180 3 pffiffiffiffiffiffiffi
GHz;
(3)
2
L w E
where is the mass density and E is the Young’s modulus
of diamond. For a beam of dimensions ðL; w; hÞ ¼
ð1; 0:1; 0:1Þ m we obtain a vibrational frequency
!m =2 1 GHz and coupling g=2 1 kHz. While
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this is smaller than the strain coupling ge =2 10 MHz
expected for electronic excited states of defect centers
[27,28] or quantum dots [29], we benefit from the much
longer spin coherence time T2 in the ground state. An
important figure of merit is the single spin cooperativity
¼ ðg2 T2 Þ=ðn th Þ, where ¼ !m =Q is the mechanical
damping rate and n th ¼ ðe@!m =kB T 1Þ1 is the equilibrium phonon occupation number at temperature T; for
example, the condition > 1 is sufficient to perform a
quantum gate between two spins mediated by a thermal
mechanical mode [10]. Assuming Q ¼ 106 , T2 ¼ 10 ms
and T ¼ 4 K, we obtain a single spin cooperativity of
0:8. This can be further increased by reducing the
dimensions of the nanobeam and operating at lower
temperatures.
Spin squeezing.—In the dispersive regime, g ¼
B !m , virtual excitations of the mechanical mode
result in effective interactions between the otherwise
decoupled spins. In this limit, H can be approximately
diagonalized by the transformation eR HeR with R ¼
g
y
2
ða J aJþ Þ. To order ðg=Þ this yields an effective
Hamiltonian,
Heff ¼ !m ay a þ ðB þ ay aÞJz þ Jþ J ;
2
(4)
where ¼ 2g2 = is the phonon-mediated spin-spin
coupling strength. Rewriting Jþ J ¼ J2 Jz2 þ Jz , and
provided the total angular momentum J is conserved, we
obtain a term / Jz2 corresponding to the one-axis twisting
Hamiltonian [30].
To generate a spin squeezed state, we initialize the
ensemble in a coherent spin state (CSS) j c 0 i along the x
axis of the collective Bloch sphere. The CSS satisfies
Jx j c 0 i ¼ Jj c 0 i and has equal transverse variances, hJy2 i ¼
hJz2 i ¼ J=2. This can be prepared using optical pumping
and microwave spin manipulation applied to the ensemble
[31]. The squeezing term / Jz2 describes a precession
of the collective spin about the z axis at a rate proportional to Jz , resulting in a shearing of the uncertainty
distribution and a reduced spin variance in one direction
as shown in Fig. 1(d). This is quantified by the squeezing
parameter [32,33],
2
2JhJmin
i
;
(5)
2
hJx i
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 i ¼ 1 ðV V 2 þ V 2 Þ is the minimum
where hJmin
þ
yz
2
spin uncertainty with V ¼ hJy2 Jz2 i and Vyz ¼ hJy Jz þ
Jz Jy i=2. The preparation of a spin squeezed state, characterized by 2 < 1, has direct implications for NV ensemble
magnetometry applications, since it would enable magnetic field sensing with a precision below the projection
noise limit [32].
We now consider spin squeezing in the presence of
realistic decoherence. In addition to the coherent dynamics
156402-2
2 ¼
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PHYSICAL REVIEW LETTERS
PRL 110, 156402 (2013)
described by Heff , we account for mechanical dissipation
and spin dephasing using a master equation [24]
Optimizing t and the detuning , we obtain the optimal
squeezing parameter,
1 X
_ ¼ i Jz2 þ ðB þ ay aÞJz ; þ
D½iz 2
2T2 i
2
2opt ’ pffiffiffiffiffiffiffi ;
J
þ ðn th þ 1ÞD½J þ n th D½Jþ ;
(6)
where D½c ¼ ccy 12 ðcy c þ cy cÞ and the single
spin dephasing T21 is assumed to be Markovian for simplicity (see below). Note that we absorbed a shift of =2
into B , and ignored single spin relaxation as T1 can be
several minutes at low temperatures [34]. The second line
describes collective spin relaxation induced by mechanical
dissipation, with ¼ g2 =2 . Finally, the phonon number n ¼ ay a shifts the spin frequency, acting as an effective fluctuating magnetic field which leads to additional
dephasing.
Let us for the moment ignore fluctuations of the phonon
number n; we address these in detail below. Starting from
the CSS j c 0 i, we plot the squeezing parameter in Fig. 2(a)
for an ensemble of N ¼ 100 spins and several values of n th ,
in the presence of dephasing T21 and collective relaxation
. Here we calculated 2 by solving Eq. (6) using an
approximate numerical approach treating and T2 separately, and verified that the approximation agrees with
exact results for small N [24]. To estimate the minimum
squeezing, we linearize the equations of motion for the
averages and variances of the collective spin operators [see
dashed lines in Fig. 2(a)]. From these linearized equations,
in the limits of interest, J 1, n th 1 and to leading order
in both sources of decoherence, we obtain approximately
2 ’
4 n th
t
þ :
T2
J2 t
1
1
0.5
0.5
0.1
0
(b)
0.2
0.4
0.6
10
pffiffiffiffiffiffiffi
at time topt ¼ T2 = J, similar to results for atomic systems
[35–37]. Note that for non-Markovian dephasing, the scaling is even more favorable [38]. In Fig. 2(b) we plot the
scaling of the squeezing parameter with J for small but finite
decoherence, and find agreement with Eq. (8). For comparison we also plot the unitary result in the absence of decoherence, scaling as 2opt J 2=3 and limited by the Bloch
sphere curvature [30].
Phonon number fluctuations.—In Eq. (4) we see that the
phonon number n ¼ ay a couples to Jz , leading to additional dephasing due to thermal number fluctuations. On
the other hand, this same coupling can also lead to additional spin squeezing from cavity feedback, by driving the
mechanical mode [35–37]. In the following, we consider
a twofold approach to mitigate thermal spin dephasing
while preserving the optimal squeezing. First, we apply a
sequence of global spin echo control pulses to suppress
dephasing from low-frequency thermal fluctuations. This
also extends the effective coherence time T2 of single NV
spins [31]. Second, we consider driving the mechanical
mode, and identify conditions when this results in a net
improvement of the squeezing.
To simultaneously account for thermal dephasing,
driven feedback squeezing, and spin control pulse sequences, we write the interaction term in Eq. (4) in the so-called
toggling frame [39],
Hint ðtÞ ¼ Jz fðtÞnðtÞ:
(7)
0.1
(a)
50
100
FIG. 2 (color online). (a) Spin squeezing parameter versus
scaled precession time with N ¼ 100 spins. Thick blue (gray)
lines show the calculated squeezing parameter for T2 ¼ 10 ms
and values of n th as shown. For each curve, we optimized the
detuning to obtain the optimal squeezing. Dashed lines are
calculated from the linearized equations for the spin operator
averages. Thin black solid (dashed) line shows exact (linearized)
unitary squeezing. (b) Optimal squeezing versus number of
spins. Lower (upper) line shows power law fit for n th ¼ 1 (10)
and T2 ¼ 1 (0.01) s. The detuning is optimized for each point.
Other parameters in both plots are !m =2 ¼ 1 GHz, g=2 ¼
1 kHz, Q ¼ 106 .
(8)
(9)
The function fðtÞ periodically inverts the sign of the interaction as shown in the inset of Fig. 3(a), describing the
inversion of the collective spin Jz ! Jz with each pulse of the spin echo sequence. Phonon number fluctua where n is the
tions are described by nðtÞ ¼ nðtÞ n,
mean phonon number and we have omitted an average
frequency shift proportional to n Rin Eq. (9). The number
fluctuation spectrum Sn ð!Þ ¼ dtei!t hnðtÞnð0Þi is
plotted in Fig. 3(a) for a driven oscillator coupled to a
thermal bath [24].
We calculate the required spin moments within the
Gaussian approximation for phonon number fluctuations,
and obtain [24]
hJþ ðtÞi ¼ e
heiðJz 1=2Þ Jþ ð0Þi;
(10)
2
ðtÞi and hJþ ðtÞJz ðtÞi. In Eq. (10)
and similar results for hJþ
the dephasing parameter and effective squeezing via are given by
156402-3
¼ 2
Z d! Fð!Þ
S ð!Þ;
2 !2 n
(11)
PRL 110, 156402 (2013)
¼ 2
PHYSICAL REVIEW LETTERS
Z d! Kð!Þ
An ð!Þ;
2 !2
(12)
where Sn ð!Þ ¼ ½Sn ð!Þ þ Sn ð!Þ=2 and An ð!Þ ¼
½Sn ð!Þ Sn ð!Þ=2. The filter function Fð!Þ ¼
R
!2
i!t
2
2 j dte fðtÞj describes the effect of the spin echo pulse
sequence with time between pulses [40–42]. The
function Kð!Þ plays the analogous role for the effective
squeezing described by , and is related to F by a
Kramers-Kronig relation [24]. We plot K and F for a
sequence of M ¼ 4 pulses in Fig. 3(a).
Discussion.—We now consider the impact of thermal
fluctuations on the achievable squeezing. The thermal
noise spectrum Sn ð!Þ ¼ 2n th ðn th þ 1Þ=ð!2 þ 2 Þ is
symmetric around ! ¼ 0. Without spin echo control
pulses, this low-frequency noise results in nonexponential
decay of the spin coherence, 0 ðtÞ ¼ 12 2 n 2th t2 (with
n th 1), familiar from single qubit decoherence [31,43].
The
pffiffiffi inhomogeneous thermal dephasing time is T2 ’
2=n th , severely limiting the possibility of spin squeezing. In particular, at time t ¼ topt we find that squeezing is
pffiffiffi
prohibited when n th > J [24]. However, one can overcome this low-frequency thermal noise using spin echo. By
applying a sequence of M equally spaced global pulses
to the spins during precession of total time t, we obtain
FIG. 3 (color online). (a) Number fluctuation spectrum of
thermal driven oscillator. Center (blue filled) peak is purely
thermal while side (green filled) peaks are due to detuned drive.
Solid (dashed) purple line shows filter function F (K) for M ¼ 4
pulses. Inset: corresponding function fðtÞ for M ¼ 4. (b) Solid
green curves show squeezing parameter versus precession time for
n th ¼ 10 and n dr ¼ 103 , 5 104 , 106 (top to bottom). Dashed
black line shows unitary squeezing. (c) Minimum squeezing
versus drive strength for n th ¼ 50, 10 (top to bottom). Symbols
mark corresponding points with (b). Dashed black line shows
unitary squeezing. Parameters in (b) and (c) are M ¼ 4, g=2 ¼
1 kHz, T2 ¼ 10 ms, N ¼ 100, !m =2 ¼ 1 GHz, Q ¼ 106 .
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th 2 n 2th t3 =M2 , suggesting that thermal dephasing can
be made negligible relative to both and T21 . For a
pffiffiffiffiffiffiffiffiffi
sufficiently large number of pulses, M n th T2 , we
recover the optimal squeezing in Eqs. (7) and (8).
Adding a mechanical drive can further enhance
squeezing via feedback; however, it also increases phonon
number fluctuations, contributing to additional dephasing.
We consider a detuned external drive of frequency !dr ¼
!m þ , leading to two additional peaks in Sn ð!Þ at
! ¼ , as shown in Fig. 3(a). The area under the left
[right] peak scales as n dr n th [n dr ðn th þ 1Þ], where n dr is the
mean phonon number due to the drive at zero temperature.
The symmetric and antisymmetric parts of this noise
contribute to dephasing and squeezing as described by
Eqs. (11) and (12). Choosing the interval t=M ¼ 2=
between pulses, we obtain additional dephasing dr ’
2
ðÞ2 n dr n th t and effective squeezing with ’ n dr t. In
the limit n dr n th , the effects of the drive dominate over
th and and we recover the ideal scaling given in
Eq. (8), even with a small number of echo pulses. This is
shown in Figs. 3(b) and 3(c) where we see that the optimal
squeezing improves with increasing n dr for a fixed number
of pulses M ¼ 4.
Finally, we discuss our assumption of uniform coupling
strength g in Eq. (2). This is an important practical issue, as
we expect the coupling to individual spins to be inhomogeneous in experiment due to the spatial variation of strain in
the beam. Nonetheless, even with nonuniform coupling,
we still obtain squeezing of a collective spin with a reduced
effective total spin Jeff < J, provided J 1. First, we note
that inhomogeneous magnetic fields resulting in nonuniform detuning are compensated by spin echo. Second, for a
distribution of coupling
gi , the effective length of
P strengths
2 P 2
g
Þ
=
g
the collective
spin
is
ð
i i for the direct squeezing
P
P i i
term, and ð i g2i Þ2 = i g4i for feedback squeezing with a
mechanical drive. Similar conclusions were reached in
atomic and nuclear systems [35–37,44]. In the case of
direct squeezing, it is important that the sign of the gi ’s
is the same to avoid cancellation; this is automatically
achieved by using NV centers implanted on the top of
the beam. For beam dimensions ð1; 0:1; 0:1Þ m analyzed
above, we estimate that N 200 NV centers can be
embedded without being perturbed by direct magnetic
dipole-dipole interactions. A reduction of the effective
spin length by factor 2 still leaves Neff 100, sufficient
to observe spin squeezing.
Conclusions.—We have shown that direct spin-phonon
coupling in diamond can be used to prepare spin squeezed
states of an NV ensemble embedded in a nanoresonator,
even in the presence of dephasing and mechanical dissipation. With further reductions in temperature, beam dimensions, and spin decoherence rates, the regime of large
single spin cooperativity 1 could be achieved. This
would allow for coherent phonon-mediated interactions
and quantum gates between two spins embedded in the
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PRL 110, 156402 (2013)
PHYSICAL REVIEW LETTERS
same resonator via Hint ¼ ðþ
1 2 þ H:c:Þ, and coupling
over larger distances by phononic channels [27].
The authors gratefully acknowledge discussions with
Shimon Kolkowitz and Quirin Unterreithmeier. This
work was supported by NSF, CUA, DARPA, NSERC,
HQOC, DOE, the Packard Foundation, the EU project
AQUTE and the Austrian Science Fund (FWF) through
SFB FOQUS and the START Grant No. Y 591-N16.
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